Updating table with max from another table

Well, I *thought* I knew my way around SQL a little bit, but I have been
beating my head on the following problem for a couple days now and I don't
have any idea where to turn next. If you could suggest any strategies or
places to look I would appreciate it very much. Thanks in advance.

Given two table defs :

create table a (
id integer,
maxtype varchar(8)
);

create table b (
id integer,
type varchar(8),
val integer
);

and data rows as follows:

select * from a;
id | maxtype
----+---------
1 |
2 |
(2 rows)

select * from b;
id | type | val
----+-------+-----
1 | type1 | 5
1 | type2 | 6
2 | type1 | 19
2 | type2 | 4
(4 rows)

And given the following task :

update a from b such that a.maxtype is set equal to the b.type whose val
number is the highest for that matching id, that is, the result :

select * from a;
id | maxtype
----+---------
1 | type2
2 | type1
(2 rows)

is to be obtained, how can this be accomplished with SQL statements? I am
looking for a single (perhaps compound ) statement to do it, no procedural
stuff